If a is a positive real number, other than 1 and a^m = x, then we write:
m = log_ax and we say that the value of log x to the base a is m.

Examples:

(i). 10³ 1000     log₁₀ 1000 = 3.

(ii). 3⁴ = 81     log₃ 81 = 4.

(iii). 2-3 =	1	log2	1	= -3.
	8		8

(iv). (.1)² = .01     log_(.1) .01 = 2.

Properties of Logarithms:

1. log_a (xy) = log_a x + log_a y

2. loga		x		= loga x - loga y
		y

3. log_x x = 1

4. log_a 1 = 0

5. log_a (xⁿ) = n(log_a x)

MIND IT !

1. Common Logarithms:

   Logarithms to the base 10 are known as common logarithms.

2. The logarithm of a number contains two parts, namely 'characteristic' and 'mantissa'.

   Characteristic: The internal part of the logarithm of a number is called its characteristic.

   Case I: When the number is greater than 1.

   In this case, the characteristic is one less than the number of digits in the left of the decimal point in the given number.

   Case II: When the number is less than 1.

   In this case, the characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and it is negative.

   Instead of -1, -2 etc. we write 1 (one bar), 2 (two bar), etc.

   Examples:-

   Number	Characteristic	Number	Characteristic
   654.24	2	0.6453	1
   26.649	1	0.06134	2
   8.3547	0	0.00123	3

   Mantissa:

   The decimal part of the logarithm of a number is known is its mantissa. For mantissa, we look through log table.